Equational Systems and Free Constructions Chung-Kil Hur Joint work with Marcelo Fiore Computer Laboratory University of Cambridge ICALP 07 12th July 2007 Chung-Kil Hur Equational Systems and Free Constructions
Contributions of the paper General abstract Definition of Equational System. 1 Development of the Theory of Equational Systems. 2 Applications of Equational Systems. 3 Chung-Kil Hur Equational Systems and Free Constructions
Overview: Definition of Equational System Equational Systems are a framework for defining models of systems. Models of Logics what we want to reason about. Models of Computational Calculi semantic domains where meanings of programs are defined. e.g. λ -calculus, π -calculus, . . . Models of Data Types semantic domains where data types and type constructors are interpreted. . . . Chung-Kil Hur Equational Systems and Free Constructions
� � Overview: Theory of Equational Systems Model ( S ) �� F ⊣ U � D Construction of free models 1 Theoretically, the models of S can be represented by a monad. Practically, it gives interesting models: it may give syntactic models. (initial algebra semantics) it may give fully abstract models. . . . Model ( S ) is cocomplete. 2 Models can be combined to form new ones (e.g. by coproducts or pushouts) in a compositional fashion. Chung-Kil Hur Equational Systems and Free Constructions
Overview: Applications of Equational System Algebraic Theories First-order equational logic Specification, correctness and implementation of abstract data types [ADJ Group ’78] Enriched Algebraic Theories [Kelly & Power ’93] Algebraic treatment of computational effects [Plotkin & Power ’03, ’04] Equational Systems Σ -monoids [Fiore, Plotkin & Turi ’99] π -algebras [Stark ’05] Nominal equational logic [Clouston & Pitts ’07] Chung-Kil Hur Equational Systems and Free Constructions
Motivation for definition: I. Signatures Signatures as Endofunctors Algebraic Theory Σ Num = { zero : 0 , succ : 1 , plus : 2 } Σ Num -algebra D ∈ Set � zero � : D 0 → D � succ � : D 1 → D � plus � : D 2 → D Equational System Σ Num ( X ) = X 0 + X 1 + X 2 on Set Σ Num -algebra D ∈ Set s : Σ Num D → D : D 0 + D 1 + D 2 → D Chung-Kil Hur Equational Systems and Free Constructions
� Motivation for definition: II. Equations Equations as parallel pairs of Functors { x, y } ⊢ plus ( succ ( x ) , y ) = succ ( plus ( x, y )) Algebraic Theory D 2 ∀ ρ : { x, y } → D 1 D �− → � zero � � � succ � � � plus � � D D D � plus ( succ ( x ) , y ) � ρ = � succ ( plus ( x, y )) � ρ ∈ D Equational System − → ( − ) 2 - Alg Σ Num - Alg − → D 2 � � succ � × id � � plus � � 1 + D + D 2 � � � � � � �− → � = D 2 [ � zero � , � succ � , � plus � ] � D � � ���� � D � � succ � � plus � � D Chung-Kil Hur Equational Systems and Free Constructions
� Motivation for definition: II. Equations Equations as parallel pairs of Functors { x, y } ⊢ plus ( succ ( x ) , y ) = succ ( plus ( x, y )) Algebraic Theory D 2 ∀ ρ : { x, y } → D 1 D �− → � zero � � � succ � � � plus � � D D D � plus ( succ ( x ) , y ) � ρ = � succ ( plus ( x, y )) � ρ ∈ D Equational System − → ( − ) 2 - Alg Σ Num - Alg − → D 2 � � succ � × id � � plus � � 1 + D + D 2 � � � � � � �− → � = D 2 [ � zero � , � succ � , � plus � ] � D � � ���� � D � � succ � � plus � � D Chung-Kil Hur Equational Systems and Free Constructions
� Motivation for definition: II. Equations Equations as parallel pairs of Functors { x, y } ⊢ plus ( succ ( x ) , y ) = succ ( plus ( x, y )) Algebraic Theory D 2 ∀ ρ : { x, y } → D 1 D �− → � zero � � � succ � � � plus � � D D D � plus ( succ ( x ) , y ) � ρ = � succ ( plus ( x, y )) � ρ ∈ D Equational System − → ( − ) 2 - Alg Σ Num - Alg − → D 2 � � succ � × id � � plus � � 1 + D + D 2 � � � � � � �− → � = D 2 [ � zero � , � succ � , � plus � ] � D � � ���� � D � � succ � � plus � � D Chung-Kil Hur Equational Systems and Free Constructions
� Motivation for definition: II. Equations Equations as parallel pairs of Functors { x, y } ⊢ plus ( succ ( x ) , y ) = succ ( plus ( x, y )) Algebraic Theory D 2 ∀ ρ : { x, y } → D 1 D �− → � zero � � � succ � � � plus � � D D D � plus ( succ ( x ) , y ) � ρ = � succ ( plus ( x, y )) � ρ ∈ D Equational System − → ( − ) 2 - Alg Σ Num - Alg − → D 2 � � succ � × id � � plus � � 1 + D + D 2 � � � � � � �− → � = D 2 [ � zero � , � succ � , � plus � ] � D � � ���� � D � � succ � � plus � � D Chung-Kil Hur Equational Systems and Free Constructions
� � � Definition of Equational System L Σ - Alg Γ - Alg � ���������� R = U Σ U Γ D Equational System T ( D ⊲ Σ ⊢ L = R : Γ) T -Algebra ( D, s : Σ D → D ) such that L ( D, s ) = R ( D, s ) Chung-Kil Hur Equational Systems and Free Constructions
� � � � Definition of Equational System L J T � T - Alg � � Σ - Alg Γ - Alg � � � ���������� R � � � = � U Σ � � U T � U Γ � D Equational System T ( D ⊲ Σ ⊢ L = R : Γ) T -Algebra ( D, s : Σ D → D ) such that L ( D, s ) = R ( D, s ) Chung-Kil Hur Equational Systems and Free Constructions
� � � � � Theorem: Basic Free Construction For T = ( D ⊲ Σ ⊢ L = R : Γ) an Equational System, K T � � � ⊥ T - Alg � � Σ - Alg D is cocomplete. � J T � � �� � Σ , Γ preserve ω -colimits. � � � U Σ F Σ ⊣ � � � � ( Σ , Γ preserve epimorphisms.) U T � � D Construction of F Σ ( V ) 0 → V + Σ 0 → V + Σ ( V + Σ0) → · · · → ( V + Σ( - )) ∗ 0 Construction of K T ( X, s : Σ X → X ) Chung-Kil Hur Equational Systems and Free Constructions
� � � � � Theorem: Basic Free Construction For T = ( D ⊲ Σ ⊢ L = R : Γ) an Equational System, K T � � � ⊥ T - Alg � � Σ - Alg D is cocomplete. � J T � � �� � Σ , Γ preserve ω -colimits. � � � U Σ F Σ ⊣ � � � � ( Σ , Γ preserve epimorphisms.) U T � � D Construction of F Σ ( V ) 0 → V + Σ 0 → V + Σ ( V + Σ0) → · · · → ( V + Σ( - )) ∗ 0 Construction of K T ( X, s : Σ X → X ) Chung-Kil Hur Equational Systems and Free Constructions
� � � � � Theorem: Basic Free Construction For T = ( D ⊲ Σ ⊢ L = R : Γ) an Equational System, K T � � � ⊥ T - Alg � � Σ - Alg D is cocomplete. � J T � � �� � Σ , Γ preserve ω -colimits. � � � U Σ F Σ ⊣ � � � � ( Σ , Γ preserve epimorphisms.) U T � � D Construction of F Σ ( V ) 0 → V + Σ 0 → V + Σ ( V + Σ0) → · · · → ( V + Σ( - )) ∗ 0 Construction of K T ( X, s : Σ X → X ) Chung-Kil Hur Equational Systems and Free Constructions
� � � � � Theorem: Basic Free Construction For T = ( D ⊲ Σ ⊢ L = R : Γ) an Equational System, K T � � � ⊥ T - Alg � � Σ - Alg D is cocomplete. � J T � � �� � Σ , Γ preserve ω -colimits. � � � U Σ F Σ ⊣ � � � � ( Σ , Γ preserve epimorphisms.) U T � � D Construction of F Σ ( V ) 0 → V + Σ 0 → V + Σ ( V + Σ0) → · · · → ( V + Σ( - )) ∗ 0 Construction of K T ( X, s : Σ X → X ) Chung-Kil Hur Equational Systems and Free Constructions
� � � � � Theorem: Basic Free Construction For T = ( D ⊲ Σ ⊢ L = R : Γ) an Equational System, K T � � � ⊥ T - Alg � � Σ - Alg D is cocomplete. � J T � � �� � Σ , Γ preserve ω -colimits. � � � U Σ F Σ ⊣ � � � � ( Σ , Γ preserve epimorphisms.) U T � � D Construction of F Σ ( V ) 0 → V + Σ 0 → V + Σ ( V + Σ0) → · · · → ( V + Σ( - )) ∗ 0 Construction of K T ( X, s : Σ X → X ) Chung-Kil Hur Equational Systems and Free Constructions
� � � � � � � � Theorem: Basic Free Construction For T = ( D ⊲ Σ ⊢ L = R : Γ) an Equational System, K T � � � ⊥ T - Alg � � Σ - Alg D is cocomplete. � J T � � �� � Σ , Γ preserve ω -colimits. � � � U Σ F Σ ⊣ � � � � ( Σ , Γ preserve epimorphisms.) U T � � D Construction of F Σ ( V ) 0 → V + Σ 0 → V + Σ ( V + Σ0) → · · · → ( V + Σ( - )) ∗ 0 Construction of K T ( X, s : Σ X → X ) Σ X s X L ( X,s ) R ( X,s ) Γ X Chung-Kil Hur Equational Systems and Free Constructions
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